4. Inyección y movimiento de paredes
5.6. Dos paredes
In the next few sections, we will move on to define each price component by a mathematical model.
5.5.1
The Intra-week Model
The high-frequency component data s0h(t) are driven by short-term physical
underlying forces during a week, they will be defined by an Intra-week Model,
denoted as ˜s0h(t). Note that the tilde ‘∼’ differentiates the model from the
The intra-week model ˜s0h(t) is of time unit one hour, and it has two com- ponents: a deterministic function and a stochastic process,
˜
s0h(t) = fh(t) + ˜xhr(t)
This deterministic function fh(t) is represented by a periodic function con-
sisting of a linear combination of sine and cosine functions of basic period 168hrs, fh(t) = α0+ 84 X m=1 αmcos 2π 168mt + 83 X m=1 βmsin 2π 168mt
The base period 168hrs is because one week has 168 hours. The deterministic
function fh(t) models the intra-day and weekday-weekend patterns of the high-
frequency data.
The stochastic process ˜xhr(t) is modeled by a mean-reversion process,
d˜xhr = −khx˜hrdt + σhd˜zh
where d˜zh = ˜εh(t)
√
dt, and ˜εh(t) is a white noise of standard Normal distribu-
tion ˜εh(t) ∼ N (0, 1) and of no temporal correlation, E {˜εh(t) · ˜εh(t + dt)} = 0.
The mean-reversion process represents the stochastic component of the high- frequency data that varies around the deterministic intra-day and weekday- weekend patterns.
The solution of the intra-week model ˜s0h(t), given its initial value at time 0, s0h0, at any future time t, refer to Appendix C.4, is
˜ s0h(t) = fh(t) + [s0h0− fh(0)] · e−kh·t+ s σ2 h 2kh (1 − e−2kh·t) · ˜ε h(t)
The solution is a Normal distribution of mean value fh(t) + [˜s0h0− fh(0)] · e−kh·t
and standard deviation q
σ2 h
2kh (1 − e
−2kh·t).
5.5.2
The Intra-year Model
The mid-frequency data are driven by mid-term seasonal forces during a year, they will be captured by an Intra-year Model, denoted as ˜s0w(t).
The intra-year model ˜s0w(t) is of time unit one week, and it again has two components: a deterministic function and a stochastic process,
˜
s0w(t) = fw(t) + ˜xwr(t)
The deterministic function fw(t) is represented by a periodic function con-
sisting of a linear combination of sine and cosine functions of basic period 52
weeks and of harmonics up to the 4th order,
fw(t) = a0+ X m=1,2,3,4 amcos( 2π 52mt) + bmsin( 2π 52mt)
The base period of 52 weeks is because one year has 52 weeks, and the har-
monics up to the 4th order is due to the spike in the frequency spectrum at
the frequency ω = 2π · 87604 , see Figure 5.4b. The deterministic function fw(t)
models the seasonal pattern of the mid-frequency data.
The stochastic process ˜xwr(t) is modeled by a mean-reversion process,
d˜xwr = −kwx˜wrdt + σwd˜zw
It represents the stochastic component of the mid-frequency data that vibrates around the deterministic seasonal pattern.
The solution of the intra-year model ˜s0w(t), given its initial value at time 0, s0w0, at any future time t, is
˜ s0w(t) = fw(t) + [s0w0− fw(0)] · e−kw·t+ s σ2 w 2kw (1 − e−2kw·t) · ˜ε w(t)
which is again a Normal distribution.
5.5.3
The Multi-year Model
The low-frequency data are driven by long-term forces that play in a timescale of many years, they will be characterized by a Multi-year Model, denoted as ˜
s0y(t).
The multi-year model ˜s0y(t) is designed on the time unit one month, and it
has two components: a deterministic function and a stochastic process ˜
The deterministic function fy(t) is simply represented by a linear trend,
fy(t) = a + b · t
which models the long-term trend of the marginal generator heat rate.
The stochastic process ˜xyr(t) is modeled by a mean-reversion process,
d˜xyr = −kyx˜yrdt + σyd˜zy
It represents the slow variations of marginal generator heat rate around its long-term trend.
The solution of the multi-year model ˜s0y(t), at any future time t, given its initial value at time 0, s0y0, is
˜ s0y(t) = fy(t) +s0y0− fy(0) · e−ky·t+ s σ2 y 2ky (1 − e−2ky·t) · ˜ε y(t)
which is also a Normal distribution.
Here we would like to discuss briefly on designing the multi-year model: the multi-year model captures the long-term dynamics of the marginal gen- erator heat rate, and this long-term dynamics is driven by grand forces like economic cycles, and cycles of generation investment and retirement. Consid- ering the complex nature of these fundamental forces, this multi-year model, which incorporates only a long-term trend and a mean-reversion process, is a simplified model; and it might only be valid when there are many historical data and used for projecting a very long time-horizon such as a few decades. If one wants to use the multi-year model for forecasting the time-horizon of a few years, one probably should replace it with a more sophisticated econo- metric model that incorporates the forecast information such as load growth, generation investment and retirement, etc. in order to have a more accurate forecast.
5.5.4
The Fuel Price Model
The fuel price model, denoted as ˜p(t), for capturing dynamics of the monthly
natural gas prices, is left undefined, because it is out of the scope of this work.
In the rest of this work, wherever the fuel price model ˜p(t) is needed, the
5.5.5
The Overall Model
So far the natural gas prices, the high-frequency component, the mid-frequency component, and the low-frequency component have been defined respectively
by a fuel price model ˜p(t), an intra-week model ˜s0h(t), an intra-year model
˜
s0w(t), and a multi-year model ˜s0y(t). Now let’s put the four models together and view them in a big picture, see Figure 5.8.
…
Hours Years Weeks I: s t′y( ) II: s tw′( ) III: s th′( )Figure 5.8: The Overall Electricity Spot Price Model - In the logarithm domain, the yearly model ˜s0y(t) lays the big skeleton, on which the weekly model ˜
s0w(t) hangs the intra-year seasonal structure, and then the hourly model ˜s0h(t) adds the intra-week variations
The overall electricity spot price model is the summation of the four sub-
models: first the multi-year model ˜s0y(t) lays the long-term variations across
multi-year model; further the intra-week model ˜s0h(t) adds the hourly variations
on the intra-year model; and finally the fuel price model ˜p(t) is constantly
adjusting the electricity spot prices. ˜
s(t) = ˜s0y(t) + ˜s0w(t) + ˜s0h(t) + ˜p(t)
Take exponential on the additive model ˜s(t), it becomes an multiplicative
model ˜S(t),
˜
S(t) = es˜0y(t)· es˜w0 (t)· e˜s0h(t)· ep(t)˜
∆
= ˜Sy0(t) · ˜Sw0 (t) · ˜Sh0(t) · ˜P (t)
Namely, the overall electricity spot price model ˜S(t) is a multiplication of four
sub-models, refer to Appendix C.5.